| // Copyright (c) 2006 Xiaogang Zhang |
| // Use, modification and distribution are subject to the |
| // Boost Software License, Version 1.0. (See accompanying file |
| // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) |
| // |
| // History: |
| // XZ wrote the original of this file as part of the Google |
| // Summer of Code 2006. JM modified it to fit into the |
| // Boost.Math conceptual framework better, and to correctly |
| // handle the p < 0 case. |
| // |
| |
| #ifndef BOOST_MATH_ELLINT_RJ_HPP |
| #define BOOST_MATH_ELLINT_RJ_HPP |
| |
| #ifdef _MSC_VER |
| #pragma once |
| #endif |
| |
| #include <boost/math/special_functions/math_fwd.hpp> |
| #include <boost/math/tools/config.hpp> |
| #include <boost/math/policies/error_handling.hpp> |
| #include <boost/math/special_functions/ellint_rc.hpp> |
| #include <boost/math/special_functions/ellint_rf.hpp> |
| |
| // Carlson's elliptic integral of the third kind |
| // R_J(x, y, z, p) = 1.5 * \int_{0}^{\infty} (t+p)^{-1} [(t+x)(t+y)(t+z)]^{-1/2} dt |
| // Carlson, Numerische Mathematik, vol 33, 1 (1979) |
| |
| namespace boost { namespace math { namespace detail{ |
| |
| template <typename T, typename Policy> |
| T ellint_rj_imp(T x, T y, T z, T p, const Policy& pol) |
| { |
| T value, u, lambda, alpha, beta, sigma, factor, tolerance; |
| T X, Y, Z, P, EA, EB, EC, E2, E3, S1, S2, S3; |
| unsigned long k; |
| |
| BOOST_MATH_STD_USING |
| using namespace boost::math::tools; |
| |
| static const char* function = "boost::math::ellint_rj<%1%>(%1%,%1%,%1%)"; |
| |
| if (x < 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "Argument x must be non-negative, but got x = %1%", x, pol); |
| } |
| if(y < 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "Argument y must be non-negative, but got y = %1%", y, pol); |
| } |
| if(z < 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "Argument z must be non-negative, but got z = %1%", z, pol); |
| } |
| if(p == 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "Argument p must not be zero, but got p = %1%", p, pol); |
| } |
| if (x + y == 0 || y + z == 0 || z + x == 0) |
| { |
| return policies::raise_domain_error<T>(function, |
| "At most one argument can be zero, " |
| "only possible result is %1%.", std::numeric_limits<T>::quiet_NaN(), pol); |
| } |
| |
| // error scales as the 6th power of tolerance |
| tolerance = pow(T(1) * tools::epsilon<T>() / 3, T(1) / 6); |
| |
| // for p < 0, the integral is singular, return Cauchy principal value |
| if (p < 0) |
| { |
| // |
| // We must ensure that (z - y) * (y - x) is positive. |
| // Since the integral is symmetrical in x, y and z |
| // we can just permute the values: |
| // |
| if(x > y) |
| std::swap(x, y); |
| if(y > z) |
| std::swap(y, z); |
| if(x > y) |
| std::swap(x, y); |
| |
| T q = -p; |
| T pmy = (z - y) * (y - x) / (y + q); // p - y |
| |
| BOOST_ASSERT(pmy >= 0); |
| |
| T p = pmy + y; |
| value = boost::math::ellint_rj(x, y, z, p, pol); |
| value *= pmy; |
| value -= 3 * boost::math::ellint_rf(x, y, z, pol); |
| value += 3 * sqrt((x * y * z) / (x * z + p * q)) * boost::math::ellint_rc(x * z + p * q, p * q, pol); |
| value /= (y + q); |
| return value; |
| } |
| |
| // duplication |
| sigma = 0; |
| factor = 1; |
| k = 1; |
| do |
| { |
| u = (x + y + z + p + p) / 5; |
| X = (u - x) / u; |
| Y = (u - y) / u; |
| Z = (u - z) / u; |
| P = (u - p) / u; |
| |
| if ((tools::max)(abs(X), abs(Y), abs(Z), abs(P)) < tolerance) |
| break; |
| |
| T sx = sqrt(x); |
| T sy = sqrt(y); |
| T sz = sqrt(z); |
| |
| lambda = sy * (sx + sz) + sz * sx; |
| alpha = p * (sx + sy + sz) + sx * sy * sz; |
| alpha *= alpha; |
| beta = p * (p + lambda) * (p + lambda); |
| sigma += factor * boost::math::ellint_rc(alpha, beta, pol); |
| factor /= 4; |
| x = (x + lambda) / 4; |
| y = (y + lambda) / 4; |
| z = (z + lambda) / 4; |
| p = (p + lambda) / 4; |
| ++k; |
| } |
| while(k < policies::get_max_series_iterations<Policy>()); |
| |
| // Check to see if we gave up too soon: |
| policies::check_series_iterations(function, k, pol); |
| |
| // Taylor series expansion to the 5th order |
| EA = X * Y + Y * Z + Z * X; |
| EB = X * Y * Z; |
| EC = P * P; |
| E2 = EA - 3 * EC; |
| E3 = EB + 2 * P * (EA - EC); |
| S1 = 1 + E2 * (E2 * T(9) / 88 - E3 * T(9) / 52 - T(3) / 14); |
| S2 = EB * (T(1) / 6 + P * (T(-6) / 22 + P * T(3) / 26)); |
| S3 = P * ((EA - EC) / 3 - P * EA * T(3) / 22); |
| value = 3 * sigma + factor * (S1 + S2 + S3) / (u * sqrt(u)); |
| |
| return value; |
| } |
| |
| } // namespace detail |
| |
| template <class T1, class T2, class T3, class T4, class Policy> |
| inline typename tools::promote_args<T1, T2, T3, T4>::type |
| ellint_rj(T1 x, T2 y, T3 z, T4 p, const Policy& pol) |
| { |
| typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type; |
| typedef typename policies::evaluation<result_type, Policy>::type value_type; |
| return policies::checked_narrowing_cast<result_type, Policy>( |
| detail::ellint_rj_imp( |
| static_cast<value_type>(x), |
| static_cast<value_type>(y), |
| static_cast<value_type>(z), |
| static_cast<value_type>(p), |
| pol), "boost::math::ellint_rj<%1%>(%1%,%1%,%1%,%1%)"); |
| } |
| |
| template <class T1, class T2, class T3, class T4> |
| inline typename tools::promote_args<T1, T2, T3, T4>::type |
| ellint_rj(T1 x, T2 y, T3 z, T4 p) |
| { |
| return ellint_rj(x, y, z, p, policies::policy<>()); |
| } |
| |
| }} // namespaces |
| |
| #endif // BOOST_MATH_ELLINT_RJ_HPP |
| |